Embodiments of the invention relate generally to diagnostic imaging and, more particularly, to a method and apparatus of iterative image reconstruction for computed tomography.
Typically, in computed tomography (CT) imaging systems, an x-ray source emits a fan-shaped beam toward a subject or object, such as a patient or a piece of luggage. Hereinafter, the terms “subject” and “object” shall include anything capable of being imaged. The beam, after being attenuated by the subject, impinges upon an array of radiation detectors. The intensity of the attenuated beam radiation received at the detector array is typically dependent upon the attenuation of the x-ray beam by the subject. Each detector element of the detector array produces a separate electrical signal indicative of the attenuated beam received by each detector element. The electrical signals are transmitted to a data processing system for analysis which ultimately produces an image.
Generally, the x-ray source and the detector array are rotated about the gantry within an imaging plane and around the subject. X-ray sources typically include x-ray tubes, which emit the x-ray beam at a focal point. X-ray detectors typically include a collimator for collimating x-ray beams received at the detector, a scintillator for converting x-rays to light energy adjacent the collimator, and photodiodes for receiving the light energy from the adjacent scintillator and producing electrical signals therefrom.
Typically, each scintillator of a scintillator array converts x-rays to light energy. Each scintillator discharges light energy to a photodiode adjacent thereto. Each photodiode detects the light energy and generates a corresponding electrical signal. The outputs of the photodiodes are then transmitted to the data processing system for image reconstruction. Alternatively, x-ray detectors may use a direct conversion detector, such as a CZT detector, in lieu of a scintillator.
CT systems typically use analytical methods such as a filtered back-projection (FBP) method to reconstruct images from the acquired projection data. FBP methods of reconstruction are based on the Fourier Slice Theorem and provide means of reconstructing an image analytically from a single pass through the acquired projection data to invert the Radon transform. The typical ramp filter in standard FBP can be modified to improve the frequency response in some situations such as high resolution imaging. These analytical algorithms allow the full range of parameter choice in filter design. If one wishes to emphasize particular spatial frequencies in the reconstructed image, the full array of established filter design techniques is available to provide whatever suppression and emphasis of varying frequencies is desired.
Alternatively, an iterative technique may be used for reconstruction to improve image quality or reduce dose, or both. For example, model-based iterative reconstruction (MBIR) methods may be used to estimate an image based on pre-determined models of the CT system, the acquired projection data, and the reconstructed image such that the reconstructed image best fits the acquired projection data.
MBIR methods are typically based on the optimization of a cost functional, which is the sum of two terms:
                                          x            ^                    =                      arg            ⁢                                                  ⁢                                          min                x                            ⁢                              {                                                                            ∑                                              m                        =                        0                                            M                                        ⁢                                                                  D                        m                                            ⁡                                              (                                                                              y                            m                                                    ,                                                                                    F                              m                                                        ⁡                                                          (                              x                              )                                                                                                      )                                                                              +                                      S                    ⁡                                          (                      x                      )                                                                      }                                                    ,                            Eqn        .                                  ⁢                  (          1          )                    where {circumflex over (x)} is is the value of x that achieves the minimum summation, ym is an integral projection measurement, Fm(x) is a forward projection function, Dm(ym,Fm(x)) is a distance measure between ym and Fm(x), and S(x) is a regularizing or penalty function. The vector x is the discretized representation of a CT image. The first term in the brackets includes modeling of the geometry of the scanner and the statistics of the measurements. Minimizing this portion of the cost functional may be considered as an attempt to reconstruct the image that most closely matches the available measurements according to a statistical metric. The second term may be considered as a penalty function that assesses costs to any portion of the image containing traits which are considered undesirable. For example, large differences between neighboring voxels are usually considered improbable in realistic imagery and can be, therefore, discouraged by S(x). MBIR reconstruction methods seek to balance the first and second terms.
In the Bayesian estimation framework, S(x) can also be viewed as the a priori probability density of the image ensemble (in which case the minimization above becomes a maximization) or, more commonly, the negative logarithm of this density. Based on this description of S(x), choosing the optimal image x includes weighting the choice by the probability that the given image would exist, independent of any measurements.
One class of a priori image models known in the art is the Markov random field (MRF). These models are characterized by the Gibbs' distribution, which expresses S(x) as the summation of instances of a potential applied to collections of neighboring voxels. This formulation of the log a priori has been shown equivalent to a voxel being independent of the entire remainder of the image when conditioned on the values of all other voxels with which it shares cliques, i.e., the Markov property. A typical expression exploiting the Gibbs' formulation is:
                                          S            ⁡                          (              x              )                                =                                    ∑                                                {                                      i                    ,                    j                                    }                                ∈                C                                      ⁢                                          b                                  i                  ,                  j                                            ⁢                              ρ                ⁡                                  (                                                                                    x                        i                                            -                                              x                        j                                                              σ                                    )                                                                    ,                            Eqn        .                                  ⁢                  (          2          )                    in which the potential ρ(•) is a non-negative, symmetric function, C is a clique of voxel pairs, and bi,j are directional weighting coefficients. In one example, cliques include voxel pairs which are nearest neighbors or are diagonally connected. Thus, a 3×3 matrix is typically used where the voxel, xi, is in the center, and the voxels, xj, surround the center voxel, xi. MRF models employing the 3×3 matrix, however, restrict control over frequency behavior of the reconstruction operator, partly due to the limited spatial support of the regularization kernel, and do not allow full control over noise and resolution detail.
Therefore, it would be desirable to design a system and method of iterative image reconstruction that overcome the aforementioned drawbacks.